What is the average rate of change of $g(x)=\dfrac{2}{x+3}$ over the interval $1\le x \le3$ ?
Explanation: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We will need to know the values of $g(1)$ and $g(3)$ to find the slope. $\begin{aligned} g(1)&=\dfrac{2}{1+3} \\\\ &=\dfrac{1}{2} \\\\\\ g(3)&=\dfrac{2}{3+3} \\\\ &=\dfrac{1}{3} \\\\\\ \dfrac{g(3)-g(1)}{3-1}&=\dfrac{\dfrac{1}{3}-\dfrac{1}{2}}{2} \\\\ &=\dfrac{\left(- \dfrac{1}{6}\right)}{2} \\\\ &=-\dfrac{1}{12} \end{aligned}$ The average rate of change of $g$ over the interval $1\le x \le3$ is $-\dfrac{1}{12}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints. ${1}$ ${2}$ ${3}$ $\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{2}$ $\frac{2}{3}$ $\frac{5}{6}$ ${1}$ $y$ $x$ $(1,g(1))$ $(3,g(3))$ secant line